Answer to Question #120643 in Algebra for anav

Question

A quadratic equation f (x) has two roots α and β . If α=-4+5i, determine the root β
and thus obtain an expression for the equation f (x)

Answer 1

If $\alpha=-4+5i$ is the root of the quadratic equation $f(x)=0,$ then $\beta=-4-5i$ is also the root of the quadratic equation $f(x)=0.$

$\alpha=-4+5i, \beta=-4-5i$

Hence

$$f(x)=a(x-\alpha)(x-\beta), a\not=0$$

$$f(x)=a(x-(-4+5i))(x-(-4-5i))$$

$$\alpha+\beta=-4+5i-4-5i=-8=-{b\over a}$$

$$\alpha\cdot\beta=(-4+5i)\cdot(-4-5i)=(-4)^2+(5)^2=41={c\over a}$$

$$f(x)=ax^2 +8ax+41a, a\not=0$$